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Differences
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Differences

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Differences[list]

gives the successive differences of elements in list.

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Differences[list,n]

gives the n^(th) differences of list.

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Differences[list,n,s]

gives the differences of elements step s apart.

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Differences[list,{n1,n2,…}]

gives the successive nk^(th) differences at level k in a nested list.

Details

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

First differences:

In[1]:=1
✖
https://wolfram.com/xid/0wgnmae-po4
Out[1]=1

Second differences:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-ooq
Out[1]=1

Scope  (3)Survey of the scope of standard use cases

Differences of rows, within each column:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-ioa
In[2]:=2
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https://wolfram.com/xid/0wgnmae-iqt

Differences of columns, within each row:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-wkr

Differences of a regularly sampled TimeSeries:

In[1]:=1
✖
https://wolfram.com/xid/0wgnmae-dpzqd7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgnmae-u5aaho
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0wgnmae-ei2yz9
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0wgnmae-dzr1nz
Out[4]=4

Compare to differencing the values:

In[5]:=5
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https://wolfram.com/xid/0wgnmae-bzdxai
Out[5]=5

Applications  (3)Sample problems that can be solved with this function

Deduce the degree of a sequence by seeing when successive differences become constant:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-ktn
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0wgnmae-mb1
Out[2]=2

Finite difference approximation to the Laplacian of a Gaussian:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-ktb
In[2]:=2
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https://wolfram.com/xid/0wgnmae-rlv
Out[2]=2

Prices of a stock at close:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-kp47u3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0wgnmae-yoz18c
Out[2]=2

Find the daily differences between stock values at close:

In[3]:=3
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https://wolfram.com/xid/0wgnmae-5n9yxs
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0wgnmae-fmmk82
Out[4]=4

Find the proportion of days with increase in price at close:

In[5]:=5
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https://wolfram.com/xid/0wgnmae-ueg4y3
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0wgnmae-or46di
Out[6]=6

Properties & Relations  (4)Properties of the function, and connections to other functions

First differences are constant for a linear function:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-efw
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0wgnmae-iyi
Out[2]=2

Second differences are constant for a quadratic function:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-ffn
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0wgnmae-cfu
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0wgnmae-rj5
Out[3]=3

Successive differences are progressively shorter:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-ps2
Out[1]=1

FoldList acts as an inverse of Differences:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-hz4
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0wgnmae-wk1
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Differences of an irregularly sampled TimeSeries may differ from differences of the values:

In[1]:=1
✖
https://wolfram.com/xid/0wgnmae-8sxf1b
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0wgnmae-o8ont5
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0wgnmae-my7jth
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0wgnmae-9fmzv3
Out[6]=6

Compare with Differences of the values:

In[7]:=7
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https://wolfram.com/xid/0wgnmae-b8ayt4
Out[7]=7

Set TemporalRegularity to True and compute differences of the new time series:

In[8]:=8
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https://wolfram.com/xid/0wgnmae-4g7lf0
In[9]:=9
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https://wolfram.com/xid/0wgnmae-75nm6o

Now the value differences agree:

In[10]:=10
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https://wolfram.com/xid/0wgnmae-nvoyti
Out[10]=10

Neat Examples  (2)Surprising or curious use cases

Successive differences mod 2 of a sequence based on the primes:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-kqc
Out[1]=1

Pattern resembling Pascal's triangle:

In[1]:=1
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https://wolfram.com/xid/0wgnmae-syi
Out[1]=1
Wolfram Research (2007), Differences, Wolfram Language function, https://reference.wolfram.com/language/ref/Differences.html (updated 2012).
✖
Wolfram Research (2007), Differences, Wolfram Language function, https://reference.wolfram.com/language/ref/Differences.html (updated 2012).

Text

Wolfram Research (2007), Differences, Wolfram Language function, https://reference.wolfram.com/language/ref/Differences.html (updated 2012).

✖
Wolfram Research (2007), Differences, Wolfram Language function, https://reference.wolfram.com/language/ref/Differences.html (updated 2012).

CMS

Wolfram Language. 2007. "Differences." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/Differences.html.

✖
Wolfram Language. 2007. "Differences." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/Differences.html.

APA

Wolfram Language. (2007). Differences. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Differences.html

✖
Wolfram Language. (2007). Differences. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Differences.html

BibTeX

@misc{reference.wolfram_2025_differences, author="Wolfram Research", title="{Differences}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/Differences.html}", note=[Accessed: 19-May-2025 ]}

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@misc{reference.wolfram_2025_differences, author="Wolfram Research", title="{Differences}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/Differences.html}", note=[Accessed: 19-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_differences, organization={Wolfram Research}, title={Differences}, year={2012}, url={https://reference.wolfram.com/language/ref/Differences.html}, note=[Accessed: 19-May-2025 ]}

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@online{reference.wolfram_2025_differences, organization={Wolfram Research}, title={Differences}, year={2012}, url={https://reference.wolfram.com/language/ref/Differences.html}, note=[Accessed: 19-May-2025 ]}